Pre-Lie Algebras and Operads in Positive Characteristic [PhD thesis]

90 Pages·2016·0.995 MB·English

by  Andrea Cesaro

No ORDRE :
E´cole doctorale r´egionale Sciences Pour l’Ing´enieur Lille Nord–de–France
Universit´e Lille 1 Sciences et Technologies
Pre-Lie Algebras and Operads in
Positive Characteristic
`
THESE
pr´esent´ee et soutenue publiquement le 13 mai 2016
pour l’obtention du
Doctorat de l’universit´e Lille 1
(sp´ecialit´e math´ematiques pures)
par
Andrea Cesaro
Composition du jury
Rapporteurs : Muriel Livernet Universit´e Paris Diderot
Fr´ed´eric Patras Universit´e de Nice Sophia-Antipolis
Examinateurs : Benoit Fresse (directeur de th`ese) Universit´e Lille 1
Antoine Touz´e Universit´e Lille 1
Joost Vercruysse Universit´e Libre de Bruxelles
Christine Vespa (co-encadrant de th`ese) Universit´e de Strasbourg
Laboratoire Paul Painlev´e
Mis en page avec la classe thloria.
Acknowledgements
The past several years I had the fortune to be guided throughout my studies in topology by
my advisor Benoit Fresse and Christine Vespa. I would like to thanks them for their essential
support, help and precious suggestions.
This thesis would never had reach its current state without the wonderful patience and
dedication of the referees Muriel Livernet and Frédréric Patras. They truly helped improving
the final version of this manuscript with their constructive remarks and by underlining the
essential elements of the thesis.
I would also take this opportunity to cheerfully thank Antoine Touzé and Joost Vercruysse
who kindly accepted to be the jury of my thesis. I am honoured by their presence.
I wish to present my gratitude to the members of the topology team from the University of
Lille1,fromwhichIhavelearnedsomuch.InparticularlyIwouldliketothankIvoDell’Ambrogio
for the long discussions we shared together. During those many occasions, he always generously
shared his mathematical views and opinions with me, and was a constant support during all
those years of the Ph.D. He was a wonderful colleague as well as an amazing friend.
IwouldliketoextendmygratitudetoallthemembersofthePaulPainlevéLaboratory:first
the staff who helps us fulfil our work on a day to days basis, then the Ph.D. students who make
the laboratory a positive and dynamic place to work. The Italian community, in particular, was
a big help to be correctly introduced and integrated to the laboratory team. My thoughts also
go to my office mates, Najib Idrissi, Landry Lavoine, Daoud Ounaissi, Sinan Yalin.
I would also like to thank the region Nord Pas-de-Calais which partially funded my Ph.D.
studies, the École Doctorale SPI and all the Université Lille 1 for welcoming me.
IthankallwhomsupportedmeduringtheseyearsandespeciallyIngridMorinforallherhelp
and support. She was amazing supporting me throughout the process of writing and correcting
the thesis and was my biggest help improving my french.
Finally, I am very grateful to my parents for their encouragements.
3
Acknowledgements
4
Contents
Acknowledgements 3
Introduction 7
1 On PreLie Algebras with Divided Symmetries 13
1.1 Operads and their monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.1 S-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.2 Operads and P-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.3 Γ(P,−) and Λ(P,−) monads. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.4 Non-symmetric operads and TP-algebras . . . . . . . . . . . . . . . . . . . . 19
1.2 On PreLie and rooted trees operads . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1 Non labelled trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.2 Labelled trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.3 The rooted trees operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 A basis of Γ(PreLie,V). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 The equivalence between ΛPreLie-algebras and p−PreLie-algebras . . . . . . . . 25
1.5 The Γ(PreLie,−) monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1 A formula for the Γ(PreLie,−) composition . . . . . . . . . . . . . . . . . . 29
1.5.2 Decompositions in corollas and normal form . . . . . . . . . . . . . . . . . . 30
1.5.3 A presentation for Γ(PreLie,−) . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.1 Brace algebras are ΓPreLie-algebras . . . . . . . . . . . . . . . . . . . . . . 37
1.6.2 Dendriform algebras are ΓPreLie-algebras . . . . . . . . . . . . . . . . . . . 39
2 Mackey Functors, Generalized Operads and Analytic Monads 41
2.1 Admissible cohomological Mackey functors on partition subgroups of the symme-
tric group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.1 Admissible cohomological Mackey functors . . . . . . . . . . . . . . . . . . . 42
2.1.2 The collection Par . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
n
2.2 The equivalence between strict polynomial functors and cohomological HPar -
n
Mackey functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1 Strict polynomial functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Cohomological HPar -Mackey functors and strict polynomial functors . . 47
n
2.3 The category ModM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
K
2.3.1 M-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5
Contents
2.3.2 The monoidal structures ⊠ and ◻. . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 The equivalence between strict analytic functors and M-modules . . . . . . . . . . 51
2.4.1 Strict analytic functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.2 The functor ev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 M-Operads and their algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.5.1 M-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.5.2 The M-operad Poly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
V
2.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6 M-PROPs and their algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.6.1 The category ModBiM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
K
2.6.2 M-PROPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.6.3 Algebras over an M-PROP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A Background 71
A.1 Operads and PROPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.1.1 Symmetric modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.1.2 Operads and their associated monads . . . . . . . . . . . . . . . . . . . . . . 74
A.1.3 PROPs and their algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2 Pre-Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.3 Strict polynomial functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.4 Polynomial Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.5 Mackey functor and cohomological Mackey functors . . . . . . . . . . . . . . . . . . 85
A.5.1 Mackey functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.5.2 Cohomological Mackey functors. . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bibliography 89
6
Introduction
The main purpose of this thesis is to study the categories of algebras over operads in the
contextofacategoryofmodulesdefinedoverafieldofpositivecharacteristic.Severalwellknown
theorems of algebraic operads which are valid over the fields of characteristic 0 fail to be true in
this more general setting.
LetKbethegroundringofourcategoryofmodules.BrieflyrecallthatanoperadP consists
of a collection {P(n)}n∈N where P(n) is a K module equipped with an action of the symmetric
grouponn-lettersS ,togetherwithcompositionproductswhichmodelthecompositionschemes
n
ofabstractoperations.ThestandardcategoryofalgebrasassociatedtoanoperadP isgoverned
by a monad on the category of K-modules denoted by S(P,−). This monad S(P,−) is given by
a generalization of the classical construction of the symmetric algebra. We explicitly have :
S(P,V)= ⊕P(n)⊗S V⊗n,
n
n∈N
foreveryK-moduleV,whereP(n)⊗S V⊗ndenotesthecoinvariantquotientofthetensorproduct
of the component P(n) of our operadnP and the tensor power V⊗n under the diagonal action of
thesymmetricgroupS .Thecompositionproductsofanoperadactuallyreflectthecomposition
n
productassociatedtoamonadofthisshape.Theclassicalcategoriesofalgebras,likenotablythe
category of commutative algebras, and the category of Lie algebras, are associated to operad.
In [Fre00] B.Fresse observe that we can associate another monad Γ(P,−) to any operad P
by replacing the coinvariants in the definition of S(P,−) by invariants. We explicitly have :
Γ(P,V)= ⊕P(n)⊗SnV⊗n,
n∈N
for every K-module V, where we use the notation ⊗Sn for this invariant construction.
Two important examples of algebraic structures come from this construction. The category
of divided power algebras is governed by Γ(Com,−), where Com is the operad of commutative
algebras. The category of p-restricted Lie algebras is governed by Γ(Lie,−), where Lie is the
operadofLiealgebras.ThemonadsS(P,−)andΓ(P,−)arerelatedbyanaturaltransformation
ofmonadstrace∶S(P,−)→Γ(P,−).Theepi-monofactorizationofthistracemapdefinesathird
interestingmonaddenotedbyΛ(P,−).Thesethreemonadscoincidewhenthecommutativering
K contains Q. But in general they are different.
For a given operad P, we have no general method to obtain a description of the structure of
analgebraoverthemonadsΛ(P,−)andΓ(P,−)intermsofgeneratingoperationsandrelations.
The first goal of my thesis is to find such presentations for a significant example of operad,
PreLie, which is associated to a category of algebras called pre-Lie algebras. Pre-Lie algebras
were introduced by M. Gerstenhaber [Ger63] in the deformation theory of associative algebras.
A pre-Lie algebra explicitly consists of a K-module V equipped with a bilinear product {−,−}
such that we have the relation :
{{x,y},z}−{x,{y,z}}={{x,z},y}−{x,{z,y}},
for every x,y,z ∈V. A pre-Lie algebra inherits a Lie bracket which is given by the commutator
of the pre-Lie product :
[x,y]={x,y}−{y,x},
for every x,y ∈ V. In [CL01] F. Chapoton and M. Livernet prove that the category of pre-Lie
algebras is associated to an operad that has an explicit description in terms of rooted trees.
7
Introduction
Examples of pre-Lie algebras notably appear in deformation theory of algebraic structures (see
[DSV15]),inoperadtheory(seeSection5.4.6[LV12]),andinrenormalizationtheoryforquantum
field theories (see [CK99]).
The main results of this thesis on pre-Lie algebras are explained in Chapter 1. First we show
that over a field of characteristic p > 0 the category of Λ(PreLie,−) algebras is isomorphic
to the category of p-restricted pre-Lie algebras introduced by A. Dzhumadil’daev in [Dzh01].
Explicitly :
Theorem A (Chapter 1, Theorem 1.4.16). We assume that the ground ring K is a field of
characteristic p. A ΛPreLie-algebra is equivalent to a K-module V equipped with an operation
{−,−}∶V ⊗V —→V satisfying the PreLie-relation and the following p-restricted PreLie-algebra
relation :
{{...{{x,y},y}...}y}={x,{...{{y,y}...}y}},
·„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„¶ ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„¶
p p
for every x,y∈V.
Then we give a presentation by generating operations and relations of the structure of an
algebra over the monad Γ(PreLie,−) which is valid over any commutative ring :
Theorem B (Chapter 1, Theorem 1.5.19). If V is a free module over the ground ring K, then
providing the module V with a ΓPreLie-algebra structure is equivalent to providing V with a
collection of polynomial maps
{−;−,...,−} ∶V ×V ×...×V —→V,
·„„„„„„„„„„„‚„„„„„„„„„„„¶ r1,...,rn ·„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„¶
n n
for all n ∈ N, where r ,...,r ∈ N and which are linear in the first variable and such that the
1 n
following relations hold :
{x;y1,...,yn}rσ(1),...,rσ(n) ={x;yσ−1(1),...,yσ−1(n)}r1,...,rn, (1)
for any σ∈S ;
n
{x;y1,...,yi−1,yi,yi+1,...,yn}r1,...,ri−1,0,ri+1,...,rn =
{x;y1,...,yi−1,yi+1,...,yn}r1,...,ri−1,ri+1,...,rn, (2)
{x;y ,...,λy ,...,y } =λri{x;y ,...,y ,...,y } , (3)
1 i n r1,...,ri,...,rn 1 i n r1,...,ri,...,rn
for any λ in K;
if yi=yi+1
{x;y1,...,yi,yi+1,...,yn}r1,...,ri,ri+1,...,rn =
(ri+rri+1){x;y1,...,yi,yi+2,...,yn}r1,...,ri+ri+1,ri+2,...,rn. (4)
i
ri
{x;y1,...,yi−1,a+b,yi+1,...,yn}r1,...,ri,...,rn = ∑{x;y1,...,a,b,...,yn}r1,...,s,ri−s,...,rn, (5)
s=0
{−;}=id, (6)
8
{{x;y ,...,y } ;z ,...,z } =
1 n r1,...,rn 1 m s1,...,sn
1
∑ {x;{y ;z ,...,z } ,...,{y ;z ,...,z } ,
si=βi+∑αi, ∏(rj!) 1 1 m α11,1,...,α1m,1 1 1 m α11,r1,...,α1m,r1
...,{yn;z1,...,zm}αn1,1,...,αnm,1,...,{yn;z1,...,zm}αn1,rn,...,αnm,rn,
z ,...,z } , (7)
1 m 1,...,1,β1,...,βm
where,togiveasensetothelatterformula,weusethatthedenominatorsr !dividethecoefficient
j
of the terms of the reduced expression which we get by applying relations (1) and (4) to simplify
terms with repeated inputs on the right hand side (see Example 1.5.11).
Itturnsoutthatsomeimportantexamplesofpre-LiealgebrahavesuchaΓ(PreLie,−)alge-
bra structure. For example the K-module ⊕ P(n) associated to an operad P is a Γ(PreLie,−)
n
algebra.
LetP beanoperad.LetV beaK-module.Byaclassicalstatementofthetheoryofoperads,
providing V with the structure of an S(P,−)-algebra amounts to giving an operad morphism
φ ∶ P → End , where End is a universal operad associated to V (the endomorphism operad
V V
of V). But we do not have an analogue of this universal operad for the study of Γ(P,−)-algebra
structures, at least if we only consider operads in the classical sense. In Chapter 2, we explain
how to define a suitable generalisation of the notion of an operad in order to work out this
problem.
The functors Sn(P,V) = P(n)⊗Sn V⊗n and Γn(P,V) = P(n)⊗Sn V⊗n which define the
summandsofthemonadsS(P,−)andΓ(P,−)associatedtoanoperadP areexamplesof(strict)
polynomial functors of degree n in the sense of Friedlander-Suslin.
In a first step we explain the definition of a category of (cohomological) Mackey functors,
whichgeneralizetheS -modulesconsideredinthedefinitionofanoperad,togetacombinatorial
n
“model” ofthecategoryofstrictpolynomialfuntorsofdegreen.Togiveanideaofourdefinition,
we consider the collection Par formed by the subgroups of the symmetric group S which are
n n
conjugate to a group of the form S ×⋯×S ≤ S where i +⋯+i = n. The cohomological
i1 ir n 1 r
Mackey functors which we consider can be defined by giving a collection of K-modules M(π),
where π ∈ Par , together with induction morphisms Indπ2 ∶ M(π ) → M(π ) and restriction
morphisms Resnπ2 ∶M(π )→M(π ) for each pair of subgrπo1up π ,π1 ∈Par s2uch that π ≤π ,
andconjugationπ1operatio2nsc ∶M(1π)→M(πσ)foreachσ∈S ,w1he2reπσ dennotesthecon1jugat2e
σ n
subgroup of π in S under the action of σ. We suppose that these operations satisfy natural
n
relations. We notably assume Indπ2Resπ2 = [π ∶ π ]Id in our category of cohomological
π1 π1 2 1 π2
Mackey functors.
We denote the category of strict polynomial functors of degree n by PolFun , and the
n
category of cohomological Mackey functors on Par by Maccoh(HPar ). We associate a strict
n n
polynomialfunctorev(M) ofdegreentoeveryobjectM ∈Maccoh(HPar )andweprovethat:
n n
Theorem C (Chapter 2, Theorem 2.2.18). Our mapping evn∶Maccoh(HParn) → PolFunn
defines an equivalence of categories from Maccoh(HPar ) the category of cohomological Mackey
n
functors on Par to the category PolFun of strict polynomial functors of degree n.
n n
Wethenconsideracategoryofanalyticfunctors,denotedbyAnFun,whoseobjectsaredirect
sums F = ⊕n∈NFn where Fn is a strict polynomial functor of degree n on the category of K-
modules. We check that the composition of functors lifts to the category AnFun, so that the
triple (AnFun,○,Id), where ○ is this composition operation and Id is the identity functor, forms
a monoidal category. We consider on the other hand a category of M-modules ModM whose
K
objects are collections M ={Mn}n∈N such that Mn ∈Maccoh(HParn), for each n. We consider
theobviousfunctorev∶ModMK →AnFunsuchthatev(M)=⊕n∈Nevn(Mn),foreveryM ∈ModMK.
Theorem C implies that this functor defines an equivalence of categories.
We make explicit a composition product ◻ and a unit object I in the category of M-modules
ModM suchthat(ModM,◻,I)formsamonoidalcategoryandweestablishthefollowingresults:
K K
Theorem D (Chapter 2, Theorem 2.4.28). The mapping ev∶M →ev(M), defines a (strongly)
monoidal functor from the category of M-modules equipped with the composition product ◻ to the
category of analytic functors AnFun equipped with the composition product ○.
9
Introduction
We then define an M-operad as a monoid object in the monoidal category of M-modules.
We denote the category of M-operads by M-Op. Theorem C and Theorem D have the following
corollary :
Corollary . The mapping P ↦ev(P) induces an equivalence of categories between the category
of M-operads and the category of analytic monads
Let us mention that our notion of an M-operad is equivalent to the notion of a Schur operad
defined in the Ph.D. thesis of Q. Xantcha [Xan10]. The main novelty of our approach is the
definition of our objects in terms of monoidal structures whereas Xantcha define his notion of
a Schur operad by using an abstract notion of polynomial operation. Xantcha’s approach is a
reminiscence of Lazard’s definition of an analizeur [Laz55].
WealreadymentionedthatthesummandsS (P,−)andΓ (P,−)ofthemonadsS(P,−)and
n n
Γ(P,−) associated to an operad P are examples of strict polynomial functors. These monads
S(P,−)andΓ(P,−)formexamplesofanalyticmonads;andsodoestheotherthirdmonadwhich
we associate to an operad Λ(P,−). We make explicit the M-operads S−(P), Γ−(P) and Λ−(P)
which correspond to these analytic monads. We prove that the category of p-restricted Poisson
algebras introduced by R. Bezrukavnikov and D. Kaled [BK08] in the study of deformation
theory of manifolds in positive characteristics is also associated to an M-operad p-Pois which is
not of this form.
To any K-module V is associated an M-operad denoted by Poly . If P is an M-operad then
V
the set of P-algebras structures on V is recovered from the set HomM-Op(P,PolyV)
Theorem E (Chapter 2, Theorem 2.5.8). Let P be an M-operad and V be a K-module. The set
of P-algebras structures on V is in bijection with HomM-Op(P,PolyV).
We also have a notion of an M-PROP which generalizes MacLane’s concept of a PROP,
and which can be used to govern categories of bialgebras. We define for instance an M-PROP
ΓBiAlg which governs the category of commutative-coassociative bialgebras with divided
Com
powers (This category of bialgebras is equivalent to the André’s category of divided power Hopf
algebras [And71] when we work in a category of connected graded K-modules).
Outlook
The work-in-progress [Ces] is devoted to the study of applications of divided symmetries
pre-Lie algebras on the theory of combinatorial Hopf algebras. J.-M. Oudom, D. Guin [OG08],
and T. Schedler [Sch13] showed that for a Lie algebra coming from a pre-Lie algebra a strongest
version of Poincaré-Birkhoff-Witt’s Theorem holds, namely the quantum PBW theorem. The
aim of this work [Ces] is to study the p-restricted case and a generalization to divided power
algebras of this result.
Plan of the thesis
The thesis is divided in three chapters.
Chapter1isdevotedtothestudyofpre-Liealgebras.WeexaminethedefinitionofΛ(PreLie,−)-
algebrasandΓ(PreLie,−)-algebrasintermsofgeneratingoperationsandrelations,andweesta-
blishtheresultsofTheoremAandTheoremB.WealsogiveabunchofexamplesofΛ(PreLie,−)-
algebra and Γ(Pre,Lie,−)-algebra structures in the concluding section of Chapter 1.
Chapter 2 is devoted to our study of the generalized operads which model the structure of
analytic monads. We explain the definition of our categories of cohomological Mackey functors
associated to our subset Par of the set of subgroups of the symmetric group S . We define
n n
the equivalence between these categories of cohomological Mackey functors and the category of
strictpolynomialfunctorsassertedinTheoremC.Thenweexplainthedefinitionofourmonoidal
structure and of our notion of operad in the category of M-modules, which fit in the results of
Theorem D and its corollary. We eventually establish the result of Theorem E and we give
examples of M-operads and of our more general notion of an M-PROP which naturally occur in
the field of algebra.
These Chapters 1 and 2 are independent articles of the author and each of these chapter
includes a self contained introduction and its own reminders.
10